Contact Bridgestone, send them a picture, and ask them if bulge represents a safety hazard. I'm going to guess they will say that it does. Make it clear to Bridgestone that you aren't asking them to cover the damage to the tire, as you have insurance for that, but they are denying coverage because they don't feel it is a safety hazard. If the manufacturer says it is a safety hazard and presents danger to you, the occupants of your vehicle and other drivers, the insurance company will have to pay.

Another option is to contact your state's Insurance Commissioner. I'm sure they'd like to hear about an insurance company that says they will have to destroy the tire to see if it is damaged, and if it isn't, tough shit for you. Here is the link to the Insurance Division in Illinois: http://www.idfpr.com/DOI/default2.asp